Falcone told Harbinger Capital Partners hedge fund investors about the accord in a letter today obtained by Bloomberg News. The companies were discussing a deal valued at as much as $20 billion, people familiar with the matter said earlier this month.

“LightSquared and Sprint will jointly develop, deploy and operate LightSquared’s 4G LTE network,” according to the letter. “Sprint will become a significant customer of LightSquared’s 4G LTE network.”

The deal is an important step for Falcone, who is gambling more than 60 percent of his hedge fund on LightSquared.

For Sprint, the third-largest U.S. wireless company, the deal would provide a new source of revenue to compete with bigger rivals.

Separately, Sprint has committed $5 billion to upgrade its network infrastructure in a project called Network Vision over the next three to five years. Sprint CEO Dan Hesse said last month that Sprint was also in talks with Clearwire on a similar network deal.

Sprint’s “Network Vision” includes phasing out its iDEN network beginning in 2013, which uses 800 MHz. Sprint’s cellular network was built around PCS frequencies (1900MHz). The heart of the project is the deployment of new multi-mode base station, which Sprint says could allow it to deploy LTE at minimal cost.

But what frequencies will Sprint use? The 1.6 GHz “L Band” seems like a bucket of hurt. Maybe that means Lightsquared would use Sprint’s (assumed) 2.6 GHz LTE for terrestrial access while Sprint would use Lightsquared’s 1.6 GHz frequencies for satellite access in areas not covered with terrestrial towers.

Sprint might be angling to build out the government’s dedicated, nation-wide, public service network. That will likely use the 700 MHz D Block and adjacent broadband and narrowband (voice) segments. Then Sprint and Lightsquared could couple that network with their shared spotbeam satellite phone service. Dual-mode satphones might be available on Sprint for both commercial (2.6GHz) and public service users (700MHz).

Perhaps the plan is to offer commercial access at 800 Mhz (Nextel), PCS and 2.6 GHz WiMAX/LTE though Sprint, with 700 MHz access for first responders. Terrestrial access at 1.6 GHz would be unnecessary in this plan. Both users could fall back to satellite access when out of terrestrial service range.

That’s just speculation, however. The Lightsquared/Sprint deal may become clearer (when) Sprint makes its LTE announcement.

Meanwhile, Charlie Ergen’s Spectacular Triple Play has apparently made him the front-runner for (useable) ATC terrestrial access. Charlie Ergen’s Dish Network is the apparent winner to buy TerreStar Networks, with a $1.375 billion cash bid. Other bidders could step forward, however, before for the June 30 auction, so it’s not a done deal yet.

This March, Dish also agreed to pay $1.4 billion for satellite provider ICO/DBSD, a similar bankrupt satphone company sharing the same 2 GHz MSS band as TerreStar. That would give Dish networks a nationwide phone network, controlling some 40 MHz of bandwidth.

Who will build the 2 GHz terrestrial infrastructure for EchoStar? That remains to be seen.